Theorem opprvalg 14374

Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	|- B = ( Base ` R )
opprval.2	|- .x. = ( .r ` R )
opprval.3	|- O = (oppr ` R )

Assertion

Ref	Expression
opprvalg	|- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

Proof of Theorem opprvalg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprval.3	. 2 |- O = (oppr ` R )
2		df-oppr 14373	. . 3 |- oppr = ( x e. _V |-> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) )
3		id 19	. . . 4 |- ( x = R -> x = R )
4		fveq2 5695	. . . . . . 7 |- ( x = R -> ( .r ` x ) = ( .r ` R ) )
5		opprval.2	. . . . . . 7 |- .x. = ( .r ` R )
6	4, 5	eqtr4di 2289	. . . . . 6 |- ( x = R -> ( .r ` x ) = .x. )
7	6	tposeqd 6519	. . . . 5 |- ( x = R -> tpos ( .r ` x ) = tpos .x. )
8	7	opeq2d 3911	. . . 4 |- ( x = R -> <. ( .r ` ndx ) , tpos ( .r ` x ) >. = <. ( .r ` ndx ) , tpos .x. >. )
9	3, 8	oveq12d 6103	. . 3 |- ( x = R -> ( x sSet <. ( .r ` ndx ) , tpos ( .r ` x ) >. ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
10		elex 2833	. . 3 |- ( R e. V -> R e. _V )
11		mulrslid 13486	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
12	11	simpri 113	. . . . 5 |- ( .r ` ndx ) e. NN
13	12	a1i 9	. . . 4 |- ( R e. V -> ( .r ` ndx ) e. NN )
14	11	slotex 13379	. . . . . 6 |- ( R e. V -> ( .r ` R ) e. _V )
15	5, 14	eqeltrid 2325	. . . . 5 |- ( R e. V -> .x. e. _V )
16		tposexg 6529	. . . . 5 |- ( .x. e. _V -> tpos .x. e. _V )
17	15, 16	syl 14	. . . 4 |- ( R e. V -> tpos .x. e. _V )
18		setsex 13384	. . . 4 |- ( ( R e. _V /\ ( .r ` ndx ) e. NN /\ tpos .x. e. _V ) -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
19	10, 13, 17, 18	syl3anc 1278	. . 3 |- ( R e. V -> ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) e. _V )
20	2, 9, 10, 19	fvmptd3 5799	. 2 |- ( R e. V -> (oppr ` R ) = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
21	1, 20	eqtrid 2283	1 |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3712   ` cfv 5377  (class class class)co 6085  tpos ctpos 6515   NNcn 9304   ndxcnx 13349   sSet csts 13350  Slot cslot 13351   Basecbs 13352   .rcmulr 13432  opp_rcoppr 14372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-sets 13359  df-mulr 13445  df-oppr 14373

This theorem is used by:  opprmulfvalg  14375  opprex  14378  opprsllem  14379